I'm working on a new build and some questions regarding concealment and the application of this particular stance have come up. I would ask them separately, but it seems to me that all of these are dependent on the interpretation of the same couple sentences of text.

If you move at least 10 feet during your turn, you gain concealment against all melee and ranged attacks until the start of your next turn. You also gain the standard benefits of concealment, but you cannot use this stance to hide in plain sight; you must still use some other terrain feature that normally allows you to use the Hide skill. The fluttering shadows make it difficult to specifically target you, but your enemies are aware of your position.

(Tome of Battle: The Book of Nine Swords, p. 76)

As I understand it, you remain in a martial stance until you declare otherwise; is this a correct assumption?

Let's say a character begins the round already in the Child of Shadow stance. They move 10+ feet and then change stance to Assassin's Stance (a swift action). Assassin's Stance state:

While you are in this stance, you gain the sneak attack ability, if you do not already have it, which deals an extra 2d6 points of damage.

(Tome of Battle: The Book of Nine Swords, p. 75)

Given my current assumptions, the character should now be able to perform a Sneak Attack. Does this sound correct?

Additionally, Child of Shadow does not seem to specify a requirement for the mode of movement; would this imply that any form of movement would apply? Most specifically I wonder about teleportation effects and flight, but I would also wonder about jump, climb, run, etc.

Regardless of sneak attack and mode of travel, does Child of Shadow imply that you continue to maintain concealment from attacks even after attacking?

Finally, and perhaps most subject to interpretation: would an obscure location count for something to aid concealment outside the context of having a tree/bush/cart/etc? An example that comes to mind is a character previously standing in front of an enemy teleports to the ceiling (using spider climb, levitation, etc.); would this be sufficient cause to consider the character hidden enough to perform a sneak attack? I presume even if this is the case, after attacking and without an object to hide behind, any additional hiding effect would be lost (although I do consider some ceilings might be low-light compared to ground level, lighting conditions are outside the scope of my curiosity here). My first thought here was that the enemy would be allowed a spot check, but I wonder what the check would be against? Since there is no hide skill in use, the typical opposed skill check seems out of place?

Edit: To expand on the answer below, it is worth remembering that there is no "facing" in D&D (must be nice to see in all direction at once :P). Just the same as teleporting from "in front of someone" to "behind someone" is impossible, teleporting "above" them doesn't mean they can't see you either.

1 Answer
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A creature is in a stance until it takes a swift action to change its stance.

While in the 1st-level Shadow Hand stance child of shadow (ToB 76) "[i]f you move at
least 10 feet during your turn, you gain concealment against all melee and ranged attacks until the start of your next turn. You also gain the standard benefits of concealment, but you cannot use this stance to hide in plain sight...."

Concealment Miss Chance
Concealment gives the subject of a successful attack a 20% chance that the attacker missed because of the concealment. If the attacker hits, the defender must make a miss chance percentile roll to avoid being struck. Multiple concealment conditions do not stack.

Concealment and Hide Checks
You can use concealment to make a Hide check. Without concealment, you usually need cover to make a Hide check.

and that's different from

Total Concealment
If you have line of effect to a target but not line of sight he is considered to have total concealment from you. You can’t attack an opponent that has total concealment, though you can attack into a square that you think he occupies. A successful attack into a square occupied by an enemy with total concealment has a 50% miss chance (instead of the normal 20% miss chance for an opponent with concealment).

The stance child of shadow doesn't give the adept total concealment or make the adept invisible; those are different effects. The stance's effect just gives him an opportunity to use the Hide skill to hide and a 20% miss chance versus melee and ranged attacks.

The DM must decide what movement the stance child of shadows requires. I would argue anytime a creature has gone from one square to a different square it has moved, but stricter interpretations--such as limiting them to actual movement modes (i.e. burrow, climb, fly, swim, and land Speed)--are equally reasonable. Jumping should qualify under either as part of a creature's land speed.

If movement places a creature in weird position, the standard means to determine if the creature has concealment apply:

To determine whether your target has concealment from your ranged attack, choose a corner of your square. If any line from this corner to any corner of the target’s square passes through a square or border that provides concealment, the target has concealment.

When making a melee attack against an adjacent target, your target has concealment if his space is entirely within an effect that grants concealment. When making a melee attack against a target that isn’t adjacent to you use the rules for determining concealment from ranged attacks.

This means while it might be weird and unexpected for the defender to find the attacker suddenly on the ceiling above him--and might entitle the attacker to a bonus to attack rolls for having higher ground--, the target will notice him unless the lines drawn don't allow it or another effect intervenes. It's D&D, and folks actually do check ceilings for enemies.

A creature then takes a swift action to change his stance from the stance child of shadow to the 3rd-level Shadow Hand stance assassin's stance (ToB 76) is eligible to inflict +2d6 sneak attack damage (PH 50).

The adept using the stance assassin's stance is eligible to inflict that sneak attack damage if he meets sneak attack's conditions: Is the target denied his dexterity bonus to Armor Class or flanked? If he is either or both, then sneak attack damage applies. As @TheXenocide noted, by itself, although counter-intuitive, neither concealment nor total concealment meet sneak attack's criteria so another method of meeting the criteria must be used. I suggest blinding the creature or turning invisible.

This question addresses using concealment or total concealment in combination with sneak attack in greater detail.